Yes on normal lenses the area in focus is nearly perfectly described as a plane and the small deviations are rarely taken into account in regular photography. There are two important factors that cause these slight imperfections: the Petzval curvature and Astigmatism. The uncorrected astigmatism is usually more severe but can be over-corrected in order to also somewhat get rid of the petzval field curvature (at the cost of sharpness). The Petzval curvature (sum) is caused only by the curvatures of the surfaces and the refractive index of the lenses involved and not the thickness of the optical system. Some optical systems such as big telescopes actually has curved image-sensor arrays to compensate for the Petzval curvature. In DSLR:s this is not an option since different lenses has different Petzval curvatures and the plane is therefore slightly curved.

When focusing and recomposing one will move the plane of focus from whatever the camera locked it's focus on. This distance will increase with the distance to the subject and increased difference in angle from where focus was acquired.

There are however special lenses where the focused area is not a plane at all. If you for example have a lens-sensor system where the lens can be tilted in one plane and the sensor in another the area of focus can be limited to a line instead.